The electron orbitals used to express the single particle density in
Equations 2.5 may be expanded in terms of any converged
basis set. In practice a plane wave basis set is used as this confers
a number of benefits.

A plane wave basis set is unbiased, it does not assume any
preconceptions of the form of the problem.

Due to Bloch's theorem plane waves are the natural choice for
the representation of electron orbitals in a periodic system.

The kinetic energy operator is diagonal in a plane wave
representation. Similarly the potential is diagonal in real space. The
use of Fast Fourier Transforms in changing between these
representations provides a large saving in computational cost (See
Section 2.5).

As a plane wave basis set is non-local no Pulay forces will
arise when calculating the forces on the ions in the system. Hence
these ionic forces may be calculated with greater efficiency. (See Section
2.6)

The principle disadvantage of a plane wave basis set is its
inefficiency. The number of basis functions needed to describe
atomic wavefunctions accurately near to a nucleus would be
prohibitive. This difficulty is overcome by the use of
Pseudopotentials to represent the potential of the ionic cores. This
approximation makes the assumption that only the valence electrons
determine the physical properties of the system. The pseudopotential
represents the potential of the nucleus and the core electrons subject
to the following conditions.

The valence wavefunction remains unchanged outside the core
region.

The pseudowavefunction within the core matches correctly at the
boundary.

The phase shift caused by the core is unchanged. (This means
that in general the pseudopotential must be different for each
angular momentum component.)

The norm of the valence wavefunction in the core is unchanged.

An example of a pseudopotential and pseudowavefunction can be seen in
Figure 2.1.

Figure 2.1: A
schematic illustration of all-electron (solid lines) and
pseudoelectron (dashed lines) potentials and their corresponding
wavefunctions. The radius at which all-electron and pseudoelectron
values match is . Source [22].